We analyze the three-dimensional shapes and kinematics of the young star cluster population forming in a high-resolution GRIFFIN project simulation of a metal-poor dwarf galaxy starburst. The star clusters, which follow a power-law mass distribution, form from the cold ISM phase with an IMF sampled with individual stars down to 4 solar masses at sub-parsec spatial resolution. Massive stars and their important feedback mechanisms are modelled in detail. The simulated clusters follow a surprisingly tight relation between the specific angular momentum and mass with indications of two sub-populations. Massive clusters (\(M_\mathrm{cl}\gtrsim 3\times 10^4 M_{\odot})\) have the highest specific angular momenta at low ellipticities (\(\epsilon\sim 0.2\)) and show alignment between their shapes and rotation. Lower mass clusters have lower specific angular momenta with larger scatter, show a broader range of elongations, and are typically misaligned indicating that they are not shaped by rotation. The most massive clusters \((M \gtrsim 10^5\,M_{\odot})\) accrete gas and proto-clusters from a \( \lesssim 100\,\rm pc\) scale local galactic environment on a \(t \lesssim 10\,\rm Myr\) timescale, inheriting the ambient angular momentum properties. Their two-dimensional kinematic maps show ordered rotation at formation, up to \(v \sim 8.5\,\rm km\, s^{-1}\), consistent with observed young massive clusters and old globular clusters, which they might evolve into. The massive clusters have angular momentum parameters \(\lambda_R\lesssim 0.5\) and show Gauss-Hermite coefficients \(h_3\) that are anti-correlated with the velocity, indicating asymmetric line-of-sight velocity distributions as a signature of a dissipative formation process.